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Properties

Label 112.5.c.a.97.1
Level $112$
Weight $5$
Character 112.97
Self dual yes
Analytic conductor $11.577$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(97,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.97"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 97.1
Character \(\chi\) \(=\) 112.97

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-49.0000 q^{7} +81.0000 q^{9} +206.000 q^{11} +734.000 q^{23} +625.000 q^{25} +1234.00 q^{29} -1294.00 q^{37} +334.000 q^{43} +2401.00 q^{49} -5582.00 q^{53} -3969.00 q^{63} -4946.00 q^{67} -2914.00 q^{71} -10094.0 q^{77} +3646.00 q^{79} +6561.00 q^{81} +16686.0 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −49.0000 −1.00000
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 206.000 1.70248 0.851240 − 0.524777i \(-0.175851\pi\)
0.851240 + 0.524777i \(0.175851\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 734.000 1.38752 0.693762 − 0.720205i \(-0.255952\pi\)
0.693762 + 0.720205i \(0.255952\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1234.00 1.46730 0.733650 − 0.679527i \(-0.237815\pi\)
0.733650 + 0.679527i \(0.237815\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1294.00 −0.945215 −0.472608 − 0.881273i \(-0.656687\pi\)
−0.472608 + 0.881273i \(0.656687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 334.000 0.180638 0.0903191 − 0.995913i \(-0.471211\pi\)
0.0903191 + 0.995913i \(0.471211\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5582.00 −1.98718 −0.993592 − 0.113026i \(-0.963946\pi\)
−0.993592 + 0.113026i \(0.963946\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −3969.00 −1.00000
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4946.00 −1.10180 −0.550902 − 0.834570i \(-0.685716\pi\)
−0.550902 + 0.834570i \(0.685716\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2914.00 −0.578060 −0.289030 − 0.957320i \(-0.593333\pi\)
−0.289030 + 0.957320i \(0.593333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10094.0 −1.70248
\(78\) 0 0
\(79\) 3646.00 0.584201 0.292101 − 0.956388i \(-0.405646\pi\)
0.292101 + 0.956388i \(0.405646\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 16686.0 1.70248
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 112.5.c.a.97.1 1
3.2 odd 2 1008.5.f.a.433.1 1
4.3 odd 2 7.5.b.a.6.1 ✓ 1
7.6 odd 2 CM 112.5.c.a.97.1 1
8.3 odd 2 448.5.c.b.321.1 1
8.5 even 2 448.5.c.a.321.1 1
12.11 even 2 63.5.d.a.55.1 1
20.3 even 4 175.5.c.a.174.1 2
20.7 even 4 175.5.c.a.174.2 2
20.19 odd 2 175.5.d.a.76.1 1
21.20 even 2 1008.5.f.a.433.1 1
28.3 even 6 49.5.d.a.19.1 2
28.11 odd 6 49.5.d.a.19.1 2
28.19 even 6 49.5.d.a.31.1 2
28.23 odd 6 49.5.d.a.31.1 2
28.27 even 2 7.5.b.a.6.1 ✓ 1
56.13 odd 2 448.5.c.a.321.1 1
56.27 even 2 448.5.c.b.321.1 1
84.83 odd 2 63.5.d.a.55.1 1
140.27 odd 4 175.5.c.a.174.2 2
140.83 odd 4 175.5.c.a.174.1 2
140.139 even 2 175.5.d.a.76.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.5.b.a.6.1 ✓ 1 4.3 odd 2
7.5.b.a.6.1 ✓ 1 28.27 even 2
49.5.d.a.19.1 2 28.3 even 6
49.5.d.a.19.1 2 28.11 odd 6
49.5.d.a.31.1 2 28.19 even 6
49.5.d.a.31.1 2 28.23 odd 6
63.5.d.a.55.1 1 12.11 even 2
63.5.d.a.55.1 1 84.83 odd 2
112.5.c.a.97.1 1 1.1 even 1 trivial
112.5.c.a.97.1 1 7.6 odd 2 CM
175.5.c.a.174.1 2 20.3 even 4
175.5.c.a.174.1 2 140.83 odd 4
175.5.c.a.174.2 2 20.7 even 4
175.5.c.a.174.2 2 140.27 odd 4
175.5.d.a.76.1 1 20.19 odd 2
175.5.d.a.76.1 1 140.139 even 2
448.5.c.a.321.1 1 8.5 even 2
448.5.c.a.321.1 1 56.13 odd 2
448.5.c.b.321.1 1 8.3 odd 2
448.5.c.b.321.1 1 56.27 even 2
1008.5.f.a.433.1 1 3.2 odd 2
1008.5.f.a.433.1 1 21.20 even 2